A New Exact Boundary at Frequency 22: Teaching and Verification Addendum for Omega Naught Class-I Icosahedral Cubature

Matthew Alan Newman

Frequency-22 cubature audit

Degree 69 positivity and degree 70/71 obstruction

This audit concerns an exact finite linear-algebra calculation on the fixed frequency-22 radial Class-I grid. Its arithmetic is in ℚ(√5). No numerical tolerance and no property of Ω₀ is used for an equality or sign decision.

Correction notice

Correction (August 27, 2026). The earlier summary selected the minimum positive degree-69 orbit after binary64 conversion and incorrectly named (22, 0, 0). Exact comparison in ℚ(√5) gives (11, 11, 0) with mass 0.003874805888537519508839054528…. The Julia verifier also contained a stale assertion and a scalar-transpose runtime error; both are corrected in this release. The corrected Python and Julia verifiers pass separately and emit the unchanged canonical degree-69 and degree-71 solution digests. Neither run is independent third-party reproduction. The 24 August 2026 research-addendum PDF is superseded by v1.1, which reports the corrected orbit, both CERTIFICATE PASS runtimes, and the unchanged theorem. The central result is unchanged: degree 69 is strictly positive, while the unique forced degree-71 solution has negative mass at (21, 1, 0).

Implementation conformance: PASS. The corrected certificate package was executed with Python 3.12.13 and Julia 1.11.6 on Linux x86-64. Both exact implementations report (11, 11, 0) as the minimum positive degree-69 orbit, (21, 1, 0) as the negative degree-71 orbit, and identical canonical solution digests. Python and Julia pass separately. This is a two-implementation conformance check, not a claim of external independent reproduction.

Source

Title: A New Exact Boundary at Frequency 22
Subtitle: Teaching and Verification Addendum for Omega Naught Class-I Icosahedral Cubature
Author: Matthew Alan Newman
Date / version: 24 August 2026; corrected 27 August 2026. Research addendum.
Relationship to Paper VI: Is supplement to 10.5281/zenodo.22068873. Additive exact calculation on the frequency-22 Class-I grid. It does not enlarge or rewrite the Paper VI theorem, whose proved range is 1 ≤ f ≤ 9.
DOI / host: No DOI is assigned to this addendum on this site. The PDF and verifiers are locally hosted in this archive.
Archival fixation: Archival fixation: locally hosted only. These same-site hashes identify the currently published bytes but do not independently timestamp or externally preserve them.
PDF: Omega_Naught_ClassI_f22_Research_Addendum_v1.1.pdfCurrent corrected addendum. 113315 bytes · SHA-256 efb7947cd9cc35a144a665672f793c8f74832b3b891d382c80f6e175ba4a8cbaSuperseded 24 August 2026 PDF: Omega_Naught_ClassI_f22_Research_Addendum_v1.0_superseded.pdf356694 bytes · SHA-256 72332ba4f97425b132fe8073aeccfd5450e19c8f62a3a654969401213dced7a1. Superseded — contains historical data later corrected. Do not cite for the current orbit. It still reports minimum orbit (22,0,0) and Python-only reproduction; retained only as provenance. Served with X-Robots-Tag: noindex, noarchive.
Python verifier (current revision r2): Download current verifier source17584 bytes · SHA-256 d9b2450ef16b138f73f89cbc1ee3a298010d27bcf1c184d5de57246a34376387. Current Python verifier is website revision r2: it refuses python3 -O, python3 -OO, and PYTHONOPTIMIZE before certificate computation. The original 16,972-byte verifier (SHA-256 ae615d26…) remains the zip and SHA256SUMS inventory member. Revision r2 does not change the certificate, the theorem, or the canonical solution digests.
Python verifier (archived original): Download original verifier source16972 bytes · SHA-256 ae615d262ecfc564aed119e5002c4fa9c4fadf35812002426298caf7c6ab89bf. Zip and SHA256SUMS inventory member. Unchanged.
Certificate package: Download all f=22 certificate files (zip)1204491 bytes · SHA-256 f63c43e1dbd1ac5b7009e66fe577b27e8a231f90fe6b01eef779147826ae55eb. ZIP bytes are a transport convenience and are excluded from the canonical payload inventory. Independent compressors are not required to reproduce identical ZIP bytes.
Canonical release candidate: RELEASE.json · RELEASE.sha256 · SHA256SUMS.txt · README_ClassI_f22.md
Inventory SHA-256 23c1d9e8f88ac98142e2809533e734c423ca508580ad3fb0ead8c0c00b1bf8d1. ZIP bytes are a transport container and are excluded from this inventory.
Julia verifier: Download Julia verifier source16723 bytes · SHA-256 5c45650b3baec9889204ac423f8ec62c9cc0782a03e5c30a8467b9cb01a1efeaPublished Julia source executed on official Julia 1.11.6; both the Python and Julia verifiers print CERTIFICATE PASS and the unchanged canonical solution digests. Runtime record: OmegaNaughtClassIF22ExactVerifier.runtime.txt.
License: Documents, certificates, JSON, and CSV: CC BY 4.0 (LICENSE). Python and Julia verifier source: MIT (LICENSE-CODE). Frozen G6_RELEASE/1.1 bytes are not rewritten; see G6_RELEASE/1.1 license clarification.

Two claims are checked on the same frozen node set: a strictly positive $H_3$ -orbit-constant rule through algebraic degree 69, and a unique invariant solution at degrees 70/71 with a negative mass on orbit $(21,1,0)$ .

Finite objects: 52 orbit representatives, the 49 × 52 moment matrix through degree 69, three exact free masses, the unique degree-70/71 solution, and an exact Farkas/cardinal dual. Arithmetic field: $\mathbb{Q}(\sqrt{5})$ .

Decisive data

Exact reported values for the frequency-22 calculation. Rationals are source data, not decimals.
Quantity	Exact reported value
Orbit representatives $\|X_{22}/H_3\|$	52
Invariant moment constraints through degree 69	49
$\operatorname{rank} A_{69}$	49
Affine solution-space dimension through degree 69	3
Free orbit $(9,7,6)$	`11607197/394254842`
Free orbit $(8,8,6)$	`2312954/161188803`
Free orbit $(8,7,7)$	`7156114/472522843`
Degree-69 positivity	Every orbit mass is strictly positive
Additional independent constraints at degrees 70/71	3
Degree-70/71 solution	Unique
Negative orbit in the forced solution	Exactly $(21,1,0)$
Exact negative mass	Sign −1 in $\mathbb{Q}(\sqrt{5})$ . Decimal orientation `-0.006609660200017031631251308016577647` is not the certificate. Download f22_negative_orbit.json.
Dual check	Exact cardinal/Farkas certificate has the corresponding negative spherical mean
Nodes	4842

Finite certificates

The path is claim → finite certificate → independent verifier. The canonical solution payloads are the exact objects hashed by the Python verifier: the SHA-256 of each payload file equals the published solution digest.

Downloadable exact certificate files derived by replaying the frozen Python verifier.
File	Role	Bytes	SHA-256
f22_degree69_solution.canonical.txt	All 52 degree-69 orbit masses; SHA-256 equals the published degree-69 digest	496324	`0865572f76c7b8c9816ce937a79361c8926abe785ac3f41b06e4f289665bf340`
f22_degree71_solution.canonical.txt	All 52 forced degree-70/71 orbit masses; SHA-256 equals the published degree-71 digest	491483	`c1e620fc9cf6d99623423cf82450540a5991679e8ded6632bd959914068949f2`
f22_degree69_orbit_masses.csv	Degree-69 masses as a CSV of exact Q(√5) components	992538	`2dcc5b274e93b78722624fe343e2f39dc5b4e3a18eadecec807272b787d32a66`
f22_degree71_orbit_masses.csv	Forced degree-70/71 masses as a CSV of exact Q(√5) components	982857	`85b8d6c7b54bea7d4cddf65282b50371d3a7be03418cf0b780df0f329e5d18b2`
f22_dual_coefficients.canonical.txt	Exact dual/Farkas coefficient vector	498013	`16b03249c777de32b9b46eb509665994d89b151e889007e8cedf9140651329fe`
f22_moment_system_spec.json	Moment-system row/column specification: representatives, pairs, free columns	4483	`5c144759dd13dfd04120f19506748bce35e364f8fa31a1e3f622997b181fcd56`
f22_negative_orbit.json	Exact negative mass at (21,1,0) and matching dual spherical mean	38906	`dc506b4ef1e216061bf602593d3c32b879c511ee413d16cf66d549913ef76be6`
SHA256SUMS.txt	SHA-256 manifest for the certificate files in this directory	682	`27386f9612590e26ab652ca23fbf773416287621d08f822b17069e5f93a4f4bd`

Positive degree-69 certificate

The fixed Class-I grid supports a strictly positive $H_3$ -orbit-constant cubature rule through algebraic degree 69. Rank 49 on 49 constraints leaves three free parameters; the displayed rational values of those free masses yield a strictly positive solution after normalization.

Exact 52-orbit weight table: f22_degree69_solution.canonical.txt. The Python reference also prints the three free masses as exact rationals and reports every orbit mass strictly positive, with the minimum at orbit $(11,11,0)$ .

Solution digest:

0865572f76c7b8c9816ce937a79361c8926abe785ac3f41b06e4f289665bf340

Degree-70/71 obstruction

Three additional independent invariant constraints at degrees 70/71 make the system square. The unique forced solution has exactly one negative orbit mass, at $(21,1,0)$ .

Reynolds averaging. For a weight assignment $Q$ on the fixed nodes, define

\overline{Q}[p]=\frac1{|H_3|}\sum_{g\in H_3}Q[p\circ g].

If $Q$ has nonnegative weights and is degree-70 exact, then $\overline{Q}$ is nonnegative, $H_3$ -orbit-constant, and degree-70 exact. The unique orbit-constant solution is negative at $(21,1,0)$ . Hence no such $Q$ exists, even when the original weights are not orbit-constant.

An independent exact dual/cardinal (Farkas) certificate has the corresponding negative spherical mean, matching the forced negative mass. Dual vector: f22_dual_coefficients.canonical.txt.

Forced-solution digest:

c1e620fc9cf6d99623423cf82450540a5991679e8ded6632bd959914068949f2

How to rerun the calculation

Download Omega_Naught_ClassI_f22_exact_verifier.r2.py (Python 3, standard library; current revision r2). No third-party packages. Exact arithmetic in $\mathbb{Q}(\sqrt{5})$ . Current Python verifier is website revision r2: it refuses python3 -O, python3 -OO, and PYTHONOPTIMIZE before certificate computation. The original 16,972-byte verifier (SHA-256 ae615d26…) remains the zip and SHA256SUMS inventory member. Revision r2 does not change the certificate, the theorem, or the canonical solution digests. The archived original remains Omega_Naught_ClassI_f22_exact_verifier.py.

python3 --version && sha256sum Omega_Naught_ClassI_f22_exact_verifier.r2.py && python3 Omega_Naught_ClassI_f22_exact_verifier.r2.py

A Julia verifier is OmegaNaughtClassIF22ExactVerifier.jl (stdlib SHA, Printf). Published Julia source executed on official Julia 1.11.6; both the Python and Julia verifiers print CERTIFICATE PASS and the unchanged canonical solution digests.

julia --version && sha256sum OmegaNaughtClassIF22ExactVerifier.jl && julia --startup-file=no OmegaNaughtClassIF22ExactVerifier.jl

Expected facts from both runs: minimum positive orbit (11, 11, 0); negative orbit (21, 1, 0); degree-69 digest 0865572f76c7b8c9816ce937a79361c8926abe785ac3f41b06e4f289665bf340; degree-71 digest c1e620fc9cf6d99623423cf82450540a5991679e8ded6632bd959914068949f2; CERTIFICATE PASS. Elapsed time is transcript metadata, not a reproducibility invariant.

Expected mathematical facts

orbit_count = 52
rank_degree_69 = 49
affine_dimension_degree_69 = 3
all_degree_69_weights_positive = true
minimum_positive_orbit = (11,11,0)
degree_71_solution_unique = true
negative_orbit = (21,1,0)
dual_certificate_valid = true

The Python program does not emit those field names. Literal stdout of the reference run follows.

OMEGA NAUGHT — EXACT CLASS-I f=22 CERTIFICATE
Arithmetic: Q(sqrt(5)); equality/sign checks: exact
nodes=4842  H3_orbits=52
degree_69_conditions=49  rank=49  affine_dimension=3
free_orbits=[(9, 7, 6), (8, 8, 6), (8, 7, 7)]
free_masses=[Fraction(11607197, 394254842), Fraction(2312954, 161188803), Fraction(7156114, 472522843)]
degree_69: EXACT and STRICTLY POSITIVE; minimum_orbit_mass=0.003874805888537520 at (11, 11, 0)
degree_69_solution_sha256=0865572f76c7b8c9816ce937a79361c8926abe785ac3f41b06e4f289665bf340
new_degree_70_invariants=[(10, 1), (5, 4), (0, 7)]
degree_70/71: UNIQUE exact solution; negative_orbit_mass=-0.006609660200017032 at (21, 1, 0)
degree_71_solution_sha256=c1e620fc9cf6d99623423cf82450540a5991679e8ded6632bd959914068949f2
Farkas/cardinal dual: EXACT PASS; mean equals negative forced mass
THEOREM: maximum nonnegative H3-orbit-constant degree on X_22 is 69
CERTIFICATE PASS

Transcript file: Omega_Naught_ClassI_f22_verification_output.txt, SHA-256 7f61a254d04afdc9e65e54355afe84ede6afba9e5874492e755758392a60bf54

Julia transcript: OmegaNaughtClassIF22ExactVerifier.runtime.txt, SHA-256 9ec4cbbc3ad02fe9291684830b669522cd4d25fdbff0d97449c3f282279a5336

What the check establishes, and what it does not

This verifier checks a specified exact finite calculation and its certificate. It is strong reproducibility evidence for that calculation. It is not peer review, and an implementation derived from the same reference source is not an independent third-party reproduction.

Implementation conformance: the named implementation reproduced the specified reference outputs. This does not establish theorem validity, peer-review status, or independent mathematical reproduction.